Totally positive matrices and totally positive hypergraphs

Linear Algebra and its Applications 331 (2001) 193–202www.elsevier.com/locate/laa

Totally positive matrices and totally positivehypergraphs

Grzegorz Kubicki a, Jen"o Lehel a,∗,1, Michał Morayne b,c,2

aDepartment of Mathematics, University of Louisville, Louisville, KY 40292, USAbInstitute of Mathematics, Wrocław University of Technology, Wybrzeze Wyspianskiego 27,

50-370 Wrocław, PolandcInstitute of Mathematics, Polish Academy of Sciences, Kopernika 18, 51-617 Wrocław, Poland

Received 16 March 2000; accepted 30 January 2001

Submitted by R.A. Brualdi

Abstract

A real matrix is totally positive if all its minors are nonnegative. In this paper, we charac-terize 0–1 matrices that can be transformed into totally positive matrices by permutations ofrows and columns. © 2001 Elsevier Science Inc. All rights reserved.

1. Introduction

A real matrix M is called totally positive of order s, TPs for short, if for every t,1 � t � s, the determinant of each t × t submatrix of M is nonnegative. A matrixwhich is TPs , for every s � 1, is called totally positive. Notice that all entries ofa totally positive matrix are nonnegative. Total positivity seems to be a significantconcept at the crossroads of analysis, statistics, and combinatorics. For the generaltheory of total positivity we recommend the monograph by Karlin [4], and we referthe reader to [9] for the combinatorial context. Certain relationship between graphsand totally positive matrices (not necessarily 0–1 matrices) is discussed in [7].

∗ Corresponding author.1 On leave from the Computer and Automation Research Institute of the Hungarian Academy of

Sciences.2 Research partially supported by KBN Grant 2P03A 01813. The paper was written when visiting the

Department of Mathematics at the University of Louisville.

0024-3795/01/$ - see front matter � 2001 Elsevier Science Inc. All rights reserved.PII: S 0 0 2 4 - 3 7 9 5 ( 0 1 ) 0 0 2 5 2 - X

194 G. Kubicki et al. / Linear Algebra and its Applications 331 (2001) 193–202

In this paper, we deal with 0–1 matrices. Our ultimate goal is a characterizationof TPs 0–1 matrices together with all those matrices which are obtained from a TPsmatrix by permutations of rows an columns. The rows of anm× n 0–1 matrix M areusually considered as characteristic vectors associated with a family E of m subsets ofan n-element set V. Using hypergraph terminology we will say that M is the incidencematrix of the hypergraph H = (V ,E). The hypergraph language used here shouldevoke Ryser’s earlier concept of combinatorial configurations in [8], notably two 0–1matrices are considered the same if their associated hypergraphs are isomorphic. Inparticular, a hypergraph is called TPs (totally positive of order s) if it is isomorphicto a hypergraph with a TPs incidence matrix.

In Section 2, we shall show how TP2 0–1 matrices are related to matrices withthe consecutive ones property, a notion introduced by Fulkerson and Gross [3] in thestudy of interval structures (see also [8]).

In Section 3, it is shown that TP2 hypergraphs form a subfamily of interval hy-pergraphs characterized by Moore [6] under the name of D-interval hypergraphs. Wealso describe TP3 hypergraphs. Perhaps unexpectedly, this completes the character-ization of TPs hypergraphs due to Theorem 2.4: If a 0–1 matrix is TP3, then it istotally positive.

In Section 4, we characterize the intersection graphs of TP2 hypergraphs and thoseof totally positive hypergraphs.

2. TPs matrices

A real sequence {an}n�0 is called logarithmically concave, or log-concave forshort, if a2

i � ai−1ai+1, for all i > 1. Sequences enumerating combinatorial objectsare usually nonnegative log-concave sequences, furthermore, they contain no internal0’s (i.e., each term is positive between any two positive terms). These combinatori-al sequences also satisfy the more general inequalities aiaj � ai−kaj+k , for everyk � 0 and k � i � j , yielding an equivalent definition of log-concavity. To see thisit is enough to write the inequalities above in an equivalent form using determinants,∣∣∣∣ aj aj+k

ai−k ai

∣∣∣∣ � 0,

and then observe that one finds all these determinants among the set of all 2 × 2subdeterminants of the infinite shift matrix of the sequence,

A =

a1 a2 a3 a4 · · · an · · ·0 a1 a2 a3 · · · an−1 · · ·0 0 a1 a2 · · · an−2 · · ··:

.

It is straightforward to check that a sequence {an}n�0 with no internal 0’s is log-concave if and only if its shift matrix A is TP2.

G. Kubicki et al. / Linear Algebra and its Applications 331 (2001) 193–202 195

A real matrix {M(i, j): 1 � i � m, 1 � j � n} is said to be monotonously con-secutive if there are positive integers p1 � p2 � · · · � pm and q1 � q2 � · · · � qmwith 1 � pi � qi � n, for i = 1, . . . ,m, such that M(i, j) /= 0 if and only if pi �j � qi . Note that if M is monotonously consecutive, then so is its transposeMT.

For the characterization of TP2 0–1 matrices we need two lemmas which arestated here in general form, for real matrices. Our first observation is that the supportof a TP2 real matrix exhibits the ‘diagonal pattern’ of a monotonously consecutivematrix. This is the content of Lemma 1 whose easy proof is omitted.

Lemma 2.1. Every TP2 real matrix with no 0 rows and columns is monotonouslyconsecutive.

As a converse to Lemma 2.1, Lemma 2.2 specifies a ‘procedure’ that creates aTP2 matrix by filling out the nonzero entries in a monotonously consecutive matrixwith appropriate terms of a log-concave sequence. The determinant of a matrix Mwill be denoted by det(M).

Lemma 2.2. Let {ak}k�1 be a sequence of real numbers. For positive integers mand n, let p1 � p2 � · · · � pm and q1 � q2 � · · · � qm be positive integers with1 � pi � qi � n, i = 1, . . . ,m, and for every 1 � i � m and 1 � j � n, define

M(i, j) ={aj if pi � j � qi,0 otherwise.

If {ak}k�1 is a log-concave sequence with no internal 0’s, then the matrix{M(i, j): 1 � i � m, 1 � j � n} is TP2.

Proof. Note that for pi = i, qi = n (i = 1, 2, . . . , n), we obtain the first m rows andn columns of the shift matrix of {ak}k�1, which is TP2, by definition. Now considerany submatrix

S =[M(i1, j1) M(i1, j2)

M(i2, j1) M(i2, j2)

],

where 1 � i1 < i2 � m and 1 � j1 < j2 � n. We shall verify that det(S) � 0.IfM(i2, j1)×M(i1, j2) = 0, then det(S) = M(i1, j1)×M(i2, j2) � 0. Assum-

ing that M(i2, j1)×M(i1, j2) /= 0, we obtain pi1 � pi2 � j1 < j2 � qi1 � qi2 .Therefore, we can writeM(i1, j1) = ax andM(i2, j2) = ay , for some pi1 � x � qi1and pi2 � y � qi2 .

If x � y, thenM(i2, j1) = ax−k, for some k � 0, and thus

S =[ax ay+kax−k ay

].

If x < y, thenM(i1, j2) = ax+�, for some � � 0, and thus

S =[ax ax+�ay−� ay

].

In both cases S is a submatrix of the shift matrix of {ak}k�1. Hence, det(S) � 0follows. �


Theorem 2.3. A 0–1 matrix with no 0 rows and columns is TP2 if and only if it ismonotonously consecutive.

Proof. Let M be a 0–1 matrix. If M is TP2 with no 0 rows and columns, then byLemma 2.1, M is monotonously consecutive. Now assume that M is monotonouslyconsecutive. Then M is TP2 due to Lemma 2.2 applied for the constant sequence(1, 1, . . .) which is obviously log-concave. �

Theorem 2.4. If a 0–1 matrix is TP3, then it is totally positive.

Proof. If a matrix is not TPs , for some s � 4, then it has a negative subdeterminant.A smallest k × k submatrix M with det(M) < 0 is clearly TPk−1, and it is not TPk ,in particular, k � 4. This matrix M has the following properties:

(1) it contains no 0 rows and no 0 columns (because det(M) < 0);

(2) it contains no identical rows or columns (because det(M) < 0);

(3) it is monotonously consecutive (by Theorem 2.3);

(4) F =1 1 0

1 1 10 1 1

is not a submatrix of M (because det(F ) = −1).

Assuming that the 0–1 matrix {M(i, j): 1 � i, j � k} satisfies properties (1)–(4)we infer the following:

(5) M(1, 1) = 1, by (1);

(6) M(1, 2) = M(2, 1) = 1, because otherwise, det(M) = det(M ′), where M ′ is a(k − 1)× (k − 1) submatrix of M, hence we would have det(M) � 0;

(7) M(2, 2) = 1, by (3).

Let p and q, 1 � p, q � k, be the smallest indices such thatM(p, 1) = 0 andM(1, q)= 0. By (5) and (6), p, q � 3. Furthermore, by (2), M(p, 2) = 1 and M(2, q) = 1.Therefore,M(p, q) = 1 follows, by (3). But the submatrix determined by row indi-ces {1, 2, p} and column indicies {1, 2, q} is identical to F contradicting (4). Hence,such M cannot exist, and this concludes the proof. �

3. TP2 hypergraphs

Let us consider the rows of an m× n 0–1 matrix M as characteristic vectors asso-ciated with a family E of m (not necessarily distinct) subsets of an n-element under-lying set V. We say that H = (V ,E) is a hypergraph with vertex set V and edge setE, and M is called the incidence matrix of H. A hypergraph H ′ corresponding to asubmatrixM ′ of M is called a subhypergraph of H. Hypergraphs associated with 0–1 matrices in this way might have isolated vertices or empty edges corresponding


to 0 column vectors or row vectors, respectively. A hypergraph with no isolatedvertices, no empty edges, and no multiple edges will be called a simple hypergraph.For further hypergraph terms, see [1].

A hypergraph is defined to be TPs if its incidence matrix has appropriate row andcolumn permutations resulting in a TPs 0–1 matrix. Notice that an arbitrary hyper-graph H is TPs if and only if the simple hypergraph obtained from H by removingall isolated vertices and empty edges, and by restricting edge multiplicities to 1, isa TPs hypergraph. The transpose of a TPs matrix is clearly TPs , by definition. Thehypergraph determined by the transpose of the incidence matrix of H is called thedual of H, and is denoted by H ∗. Hence, if H is a TPs hypergraph, then its dual H ∗is also TPs . A hypergraph which is TPs , for every s � 1, is called totally positive.Theorem 2.4 has the following form in terms of hypergraphs.

Theorem 3.1. Every TP3 hypergraph is totally positive.

A 0–1 matrix has the consecutive ones property for rows (for columns) if everyrow (every column) of M is a sequence with no internal 0’s. A hypergraph H =(V ,E) is called an interval hypergraph if there is a path P with vertex set V suchthat, for every edge ei ∈ E, the vertices of ei induce a subpath Pi ⊆ P . The sub-paths of P can be considered as discrete intervals, and they will be referred as to aninterval representation of H. By definition, H is an interval hypergraph if and onlyif its vertices have an ordering such that the corresponding incidence matrix has theconsecutive ones property for rows. Theorem 2.3 has the immediate corollary thatthe family of TP2 hypergraphs is a ‘self-dual’ restriction of the family of intervalhypergraphs: if H is a TP2 hypergraph, then its dual H ∗ is also TP2, and both H andH ∗ are interval hypergraphs.

We say that an interval I strictly contains an interval J, provided J ⊂ I , moreovertheir endpoints are different. An interval hypergraph is called an aligned intervalhypergraph if it has an interval representation with no two intervals one strictly con-taining the other. The difference hypergraph D(H) associated with H = (V ,E) isformed by adding to E all nonempty sets of the form e\f , for e, f ∈ E. Moore in[6] defines a hypergraph to be a D-interval hypergraph if the difference hypergraphD(H) of H is an interval hypergraph. The following result determines the niche ofTP2 hypergraphs in the hierarchy of interval hypergraphs.

Theorem 3.2. For any hypergraph H the following statements are equivalent:(1) H is a TP2 hypergraph;

(2) H is an aligned interval hypergraph;

(3) H is a D-interval hypergraph.

Proof. Without loss of generality, we will assume in the proof that H is a simplehypergraph.


(1) ⇐⇒ (2) Let H be a TP2 hypergraph, and let M be its TP2 incidence matrix.Because H is simple, M is monotonously consecutive, by Theorem 2.3. In particular,M has the consecutive ones property for rows, thus M obviously yields an alignedinterval representation of H.

Now assume that H has an aligned interval representation with a set{e1, e2, . . . , em} of subpaths (discrete intervals) of some underlying path(v1, v2, . . . , vn). Let M be the incidence matrix of H where columns are indexedby vi , i = 1, . . . , n, and rows are indexed by ei , i = 1, . . . ,m, as follows. Forei = {vk, . . . , v�}, define pi = k and qi = �, and assume that i < j implies pi � pj ,and qi < qj is true whenever pi = pj . This matrix is obviously monotonouslyconsecutive, hence it is TP2, by Theorem 2.3.

(2) ⇐⇒ (3) If H is an aligned interval hypergraph, then in an aligned interval rep-resentation of H any set of the form e\f is a subpath. Now assume that H is a D-inter-val hypergraph, and consider an interval representation of its difference hypergraphD(H). Let e and f be two edges such that f ⊂ e. Because e = f ∪ (e\f ) and e, f ,and (e\f ) are subpaths, e and f share a common endpoint. Thus, the representationis an aligned interval representation for H. �

Proposition 3.3. The interval hypergraphs,H0, H1, H2, H∗1 and H ∗

2 in Fig. 1, arenot TP2.

Proof. It is enough to show that these hypergraphs are not aligned interval hyper-graphs, then the claim follows, by Theorem 3.2.

InH0, one of the three singletons is not an endpoint of the 3-element edge. InH1,the singleton edge e will be strictly contained in the 3-element edge f. Indeed, for anypermutation of the three vertices that do not ‘disconnect’ the other two 2-elementedges, e is going to be the middle vertex of f. Similarly, in H2, one of the singletonedges is not the endpoint of the 3-element edge; in H ∗

1 , one of the 2-element edgeswill be strictly contained in the 4-element edge; in H ∗

2 , the singleton edge will bestrictly contained by one of the 3-element edges. �

Moore [6] characterized interval hypergraphs and D-interval hypergraphs in termsof forbidden subhypergraphs. Moore proved that the five hypergraphs in Proposi-tion 3.3 are the only minimal interval subhypergraphs forbidden in D-interval hyper-graphs. Moore’s result together with Theorem 3.2 gives a forbidden subhypergraphcharacterization of TP2 hypergraphs as follows.

Theorem 3.4. An interval hypergraph is TP2 if and only if it contains no subhyper-graph isomorphic to H0, H1, H2, H

∗1 , and H ∗

2 in Fig. 1.

It is easy to check that the only TP2 0–1 matrix of size 3 × 3 which has nega-tive determinant is a row and column permutation of the incidence matrix of H3 in


Fig. 1. Minimal not totally positive interval hypergraphs.

Fig. 1 (also called F in the proof of Theorem 2.4). Thus, Theorems 3.1 and 3.4 havethe following corollary.

Theorem 3.5. An interval hypergraph is totally positive if and only if it contains nosubhypergraph isomorphic to H0, H1, H2, H

∗1 , H

∗2 , and H3 in Fig. 1.

To see that the dual of a totally positive hypergraph or a TP2 hypergraph inheritsthe same property, let us notice that the dual of every forbidden subhypergraph isalso forbidden, because H ∗

3 = H3, furthermoreH ∗0 is not an interval hypergraph.

4. Line graphs of totally positive hypergraphs

The line graph of a hypergraphH = (V ,E) is a graph G with vertex set E, ande, f ∈ E being adjacent in G if and only if e ∩ f /= ∅. The line graph of an in-terval hypergraph is called an interval graph. Claw-free interval graphs are calledproper interval graphs due to the fact that they are the line graphs of Sperner inter-val hypergraphs (Sperner hypergraphs have no two edges one containing the other).For further terms in intersection graph theory, see [5]. In a recent paper, Deogunand Gopalakrishnan [2] characterize proper interval graphs in terms of a ‘double’


consecutive ones property of the clique matrix. Here we determine the line graph ofTP2 hypergraphs and the more restricted family of totally positive hypergraphs.

Note that Sperner interval hypergraphs are clearly aligned interval hypergraphs(no containment—nothing to be aligned). Thus, Sperner interval hypergraphs areTP2 due to Theorem 3.2. Moreover, the class of TP2 hypergraphs is obviously larg-er than that of the Sperner interval hypergraphs. However, as the next observationshows, the two hypergraph classes do not differ regarding their line graphs.

Proposition 4.1. The line graph of any TP2 hypergraph is a proper interval graph(i.e., a claw-free interval graph). Furthermore, every proper interval graph is theline graph of some TP2 hypergraph.

Proof. Let H be a TP2 hypergraph, and let G be its line graph. Every TP2 hyper-graph is an interval hypergraph, by Theorem 3.2, hence G is an interval graph. Tosee that G does not contain the 3-star K1,3 (called claw), notice that each of thethree intervals representing the three leaves must contain a point from the fourthinterval representing the center. These three points define hypergraph H0 in Fig. 1.By Theorem 3.4,H0 is forbidden in H, hence G is claw-free.

For the second part, let G be a proper interval graph, and let H be an arbitrarySperner interval hypergraph whose line graph is G. Clearly, H is an aligned hyper-graph, thus it is TP2, by Theorem 3.2. �

Let C1, C2, . . . , Cn be a sequence of cliques such that, for every 1 � i < j � n,V (Ci) ∩ V (Cj ) /= ∅ if and only if |i − j | = 1, furthermore V (Ci) does not containV (Cj ), and V (Cj ) does not contain V (Ci). Then the graphG = C1 ∪ C2 ∪ · · · ∪ Cnis called an elementary chain of cliques.

Theorem 4.2. A graph G is the line graph of some totally positive hypergraph ifand only if G is an elementary chain of cliques.

Proof. Assume thatG = C1 ∪ C2 ∪ · · · ∪ Cn is an elementary chain of cliques, andlet G contain m vertices. Let M be the m× n vertex/clique incidence matrix of G,where the columns of M are indexed with C1, C2, . . . , Cn, and the rows of M areindexed by the vertices of G in the order they occur in cliques with increasing in-dices. Then each row contains either one nonzero entry or two 1’s in consecutivepositions. Because M is obviously monotonously consecutive, it is TP2, by Theorem2.3. Furthermore, M has no three nonzero entries in one row, hence it does not containa submatrix which has row and column permutations taking it into F (the incidencematrix of the hypergraphH3 in Fig. 1). Hence, M is totally positive, by Theorem 3.5.Let H be the hypergraph associated with the incidence matrix M (rows correspondingto the edges and columns corresponding to the vertices of the hypergraph, as usual).Because each pair of intersecting edges of H corresponds to two vertices of G in the


same clique of the chain, furthermore any edge of G is covered by some clique ofthe chain, it follows that the line graph of H is isomorphic to G.

To verify the converse assertion, let H = (V ,E) be a totally positive hypergraphwith n vertices and m hyperedges. Let G be the line graph of H. By Proposition 4.1,G is a claw-free interval graph. A simple graph with four vertices and five edges willbe called a diamond.

Case 1. G is diamond-free (it has no induced subgraph isomorphic to a diamond). Inany diamond-free interval graph each maximal connected subgraph of G with no cutvertex (i.e., each block) must be a clique. Because G is claw-free, every vertex of G isincident with at most two maximal cliques of G. Moreover, every maximal cliquecontains at most two cut vertices of G, because otherwise, G would contain a triangleinduced by three cut vertices in one block and one edge at each vertex going out fromthis block, but the triangle together with the three pendant edges is clearly not aninterval graph. Thus, we conclude that G is an elementary chain of its maximal cliques.

Case 2. G contains a diamond. Let us define an equivalence relation ∼G on thevertex set of G as follows. For e, f ∈ E, e ∼G f if and only if e and f are adjacentin G, morever the set of all other vertices of G adjacent to e and the set of all othervertices of G adjacent to f are identical.

Let M be the m× n incidence matrix of H, and assume, w.l.o.g., that M is totallypositive. Let M ′ be the 4 × n submatrix of M corresponding to a diamond of G.Assume that the rows ofM ′ are indexed with e1, e2, e3, e4 ∈ E, and let pe1 � pe2 �pe3 � pe4 and qe1 � qe2 � qe3 � qe4 such that pei � qei and M ′(i, j) = 1 if andonly if pei � j � qei , for 1 � i � 4 and 1 � j � n.

Observe that (e1, e4)must be the nonadjacent pair of G, that is qe1 < pe4 , further-more pe2 � pe3 � qe1 < pe4 � qe2 � qe3 . Supposing that pe2 < pe3 , the submatrixformed by the rows e1, e2, e3 and by the columns pe2, qe1, pe4 is F, a contradiction.Thus, pe2 = pe3 , and a similar argument shows that qe2 = qe3 . We conclude that e2and e3 are identical hyperedges of H, therefore we obtain e2 ∼G e3.

Let G0 be a subgraph of G containing one vertex from each equivalence class ofG. The argument above shows that G0 is diamond-free. Thus, by Case 1, G0 is anelementary chain of cliques. Observe that the addition of an equivalent vertex to anelementary chain of cliques results in an elementary chain of cliques. Starting withG0 and by successively adding equivalent vertices to it we obtain G. Hence, G is anelementary chain of cliques. �

References

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(1999) 15–20.


[3] D.R. Fulkerson, O.A. Gross, Incidence matrices and interval graphs, Pacific J. Math. 15 (1965) 835–855.

[4] S. Karlin, Total Positivity. Vol. I, Stanford University Press, 1968.[5] T.A. McKee, F.R. McMorris, Topics in Intersection Graph Theory, SIAM Monographs on Discrete

Mathematics and Applications, 1999.[6] J.I. Moore Jr., Interval hypergraphs and D-interval hypergraphs, Discrete Math. 17 (1977) 173–179.[7] J.M. Peña, On the relationship between graphs and totally positive matrices, SIAM J. Matrix Anal.

Appl. 19 (1998) 369–377.[8] H.J. Ryser, Combinatorial configurations, SIAM J. Appl. Math. 17 (1969) 593–602.[9] R.P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, New

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Totally positive matrices and totally positive hypergraphs